Gradient flow of the Chapman–Rubinstein–Schatzman model for signed vortices
Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 2, pp. 217-246.

We continue the study of Ambrosio and Serfaty (2008) [4] on the Chapman–Rubinstein–Schatzman–E evolution model for superconductivity, viewed as a gradient flow on the space of measures equipped with the quadratic Wasserstein structure. In Ambrosio and Serfaty (2008) [4] we considered the case of positive (probability) measures, while here we consider general real measures, as in the physical model. Understanding the evolution as a gradient flow in this context gives rise to several new questions, in particular how to define a “Wasserstein” distance for signed measures. We generalize the minimizing movement scheme of Ambrosio et al. (2005) [3] in this context, we show the entropy argument of Ambrosio and Serfaty (2008) [4] still carries through, and derive an evolution equation for the measure which contains an error term compared to the Chapman–Rubinstein–Schatzman–E model. Moreover, we also show the same applies to a very similar dissipative model on the whole plane.

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author = {Ambrosio, Luigi and Mainini, Edoardo and Serfaty, Sylvia},
title = {Gradient flow of the {Chapman{\textendash}Rubinstein{\textendash}Schatzman} model for signed vortices},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {217--246},
publisher = {Elsevier},
volume = {28},
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Ambrosio, Luigi; Mainini, Edoardo; Serfaty, Sylvia. Gradient flow of the Chapman–Rubinstein–Schatzman model for signed vortices. Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 2, pp. 217-246. doi : 10.1016/j.anihpc.2010.11.006. http://archive.numdam.org/articles/10.1016/j.anihpc.2010.11.006/

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